220 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
[406 
whence, as stated, the general solution is Particular Solution 4- Constant. In the case 
in hand, taking successively (3Z) = (:/), (•//), and (///), we have in the first of 
these cases 
(I> l)w+wi' (1> l)m = ^ "t 2l)l, 
whence (I, 1)™ = n + 2m + const. = (I, !)(.'.); and the value of the constant being in 
any way ascertained to be = — 2, we have (I, l)(.‘.) = n+2m—2; and the like for the 
other three cases. 
65. (3°) The expressions for the number of conics which satisfy such conditions 
as (1), (2), &c. are deducible with more or less facility from the corresponding 
expressions wherein (1), (2), &c. are replaced by (•), (:), &c.; thus from (*)(:: l) = n + 2m 
we deduce 
(.*. I, 1) = (::/)- 2 (/. 2) = n + 2m - 2, 
viz. if one of the four arbitrary points of (::/) becomes a point on the curve, then the 
condition (:: /) is satisfied specially by the conic (.\ 2) which passes through the 
remaining three points and touches the curve at the point in question; 2 of the conics 
{::/) coincide with the conic in question. We have thus a reduction 2 (.'.2), =2, and 
the number of the conics (.*. 1, 1) is =n+2m— 2. Similarly, we have the system 
(.*.1,1 ) = n + 2 in — 2, 
( : T, T, 1 ) = n + 2m- 4, 
( • T, T, T, 1) = n + 2m — 6, 
(I, 1, 1, 1, 1) — n + 2m — 8. 
Again, two or even three of the given points on the curve may come together without 
any reduction being thereby caused, that is, we have 
(: 2, 1 ) =n + 2m — 4, 
(• 2, I, 1 ) = ( • 3, 1 ) = n + 2m - 6, 
( 2, I, I, 1) = ( 3, I, l) = w + 2m— 8 ; 
but if the four points on the curve coincide in pairs, or, what is the same thing, if 
in (2, 1, 1, 1) the points 1 and 1 come to coincide, then there is a special reduction, 
and we have 
(2, 2, 1) = n + 2m — 8 [ — (m— 2)] = m + n — 0, 
viz. here (in —2) of the conics come to coincide with the two points considered as a 
point-pair or infinitely thin conic. If the points 2 and 2 come to coincide, that is, if 
the four given points on the curve all coincide, there is no further reduction, but we 
have 
(4, 1) = m 4- n — 6. 
1 I write indifferently (!)(::), (1::) or (:: 1); and so in other cases.